Former print media geek turned video geek.
Its great to see these devices evolving! Now if we could just get the FAA to evolve a bit so we can fly with less restrictions, life would be good...very good.
Keeping focus with a shallow depth of field is tough, so for those shots consider a deeper DOF. Shooting with a wide lens can also help. Of course, these options may not give you the look you are seeking.
+1 for the Oscar Sound Tech lav mics. Very affordable, and they are a noticeable improvement over the stock lav mics that come with the Sennheiser G3 wireless system.
Lots of great answers above. I will just add to be sure you aren't panning too quickly. This can contribute to a lot of jerkiness that no stabilizer rig can fix.
Double check the settings on the monitor to see if there is an option to match frame rate to the camera. The GH3 and GH4 cameras are finicky about their HDMI output. As dj73 mentions, switching between MOV and AVCHD sometimes makes the handshake happen, but I have never been able to send 30fps via MOV out of the GH4's HDMI port.
Guy - Very nice audio quality.